Partial geometric designs having circulant concurrence matrices

We survey partial geometric designs and investigate their concurrences of points. It is well-known that the concurrence matrix of a partial geometric design can have at most three distinct eigenvalues, all of which are non-negative integers. The concurrence type of a design gives a simple profile of the incidence structure of the design. Namely, an ordinary 2-(v, k, λ) design has a single concurrence λ, and its concurrence matrix is circulant, a partial geometry has two concurrences 1 and 0, and a transversal design TDλ(k, u) has two concurrences λ and 0. In this paper, we show the existence of other partial geometric designs having two or three concurrences, and investigate which symmetric circulant matrices are realized as the concurrence matrices of partial geometric designs. We collect known sources of partial geometric designs and study their structural characteristics and construction methods. We give a list of partial geometric designs of order up to 12 each of which has a circulant concurrence matrix. We then classify these partial geometric designs along with their combinatorial properties and constructions.1